On the uniqueness of classical solutions of Cauchy problems

Bayraktar, Erhan; Xing, Hao

doi:10.1090/s0002-9939-10-10306-2

Cited by 18 publications

(29 citation statements)

References 14 publications

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“…This, together with the aforementioned non-standard Feynman-Kac formula, characterizes U as the unique classical solution to (1.4), under local Hölder continuity of σ with exponent δ ∈ (0, 1] and (1.5) (Theorem 4.5). This in particular generalizes [4,Theorem 2].…”

Section: Introductionsupporting

confidence: 78%

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The stochastic solution to a Cauchy problem for degenerate parabolic equations

Chen

Huang

Song

et al. 2017

Journal of Mathematical Analysis and Applications

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We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ ∈ (0, 1], the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement δ ≥ 1/2. Uniqueness results, including a FeynmanKac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.

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Section: Introductionsupporting

confidence: 78%

“…This result is non-standard, compared to [11,Chapter 6] and [17,Section 5.7], in that under (1.5) σ may grow faster than linearly and there is no continuity assumption on σ. Moreover, this Feynman-Kac formula generalizes Bayraktar & Xing [4,Theorem 1]; see Remark 3.16.…”

Section: Introductionmentioning

confidence: 56%

The stochastic solution to a Cauchy problem for degenerate parabolic equations

Chen

Huang

Song

et al. 2017

Journal of Mathematical Analysis and Applications

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“…Bayraktar and Xing [5] characterize one-dimensional time-homogeneous Cauchy problems with a unique solution. For the relevance of super-and sub-solutions in the study of partial differential equations of parabolic type we refer to the recent paper by [4] and the references therein.…”

Section: Proposition 55 (Unique Bounded Solution)mentioning

confidence: 99%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

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We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which To the Memory of Marc Yor. for their insightful remarks that improved the manuscript. We thank Alex Mijatović for raising the issues of strict positivity and full support and for prompting us to think about them (Sects. 4.2 and 4.3, respectively). We are grateful to the referees for their meticulous reading of the paper and their many and incisive suggestions. J.R. acknowledges generous support from the Oxford-Man Institute of Quantitative Finance, University of Oxford, where a major part of this work was completed. The problem discussed here was suggested to us by Marc Yor, who also gave us generous expert advice on several of its aspects. We dedicate the paper to his memory with deep sorrow at his passing. I.

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“…The estimation (2.4) also implies that V ∈ C γ (Q). For the necessary and sufficient condition on uniqueness, we refer the proof to Bayraktar and Xing (2010). It remains to show V is the smallest lower bounded solution.…”

Section: Characterization Of the Option Price Vmentioning

confidence: 99%

“…The necessary and sufficient condition for the unique solvability of the Black-Scholes PDE is that the underlying stock price is a true martingale process, see Bayraktar and Xing (2010). In other words, if the stock price is a strict local martingale, then there exist multiple solutions for the Black-Scholes PDE.…”

Section: Introductionmentioning

confidence: 99%

Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution

Song

Yang

2013

Quantitative Finance

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When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, the Black-Scholes PDE associated with a European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knock-out barrier moves to infinity under suitable conditions.

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On the uniqueness of classical solutions of Cauchy problems

Cited by 18 publications

References 14 publications

The stochastic solution to a Cauchy problem for degenerate parabolic equations

The stochastic solution to a Cauchy problem for degenerate parabolic equations

Distribution of the time to explosion for one-dimensional diffusions

Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution

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